序贯高斯模拟在矿石品位估计中的应用研究

随机模拟是地质统计方法的重要内容。在矿石品位估计方法中克里格方法作为一种无偏估计方法,常被用于矿石品位的估计。但克里格法估值存在平滑效应。作者在分析序贯高斯模拟和普通克里格法基本原理的基础上,运用序贯高斯模拟方法和普通克里格方法对某铁矿体内全铁(TFe)品位进行估计,给出了品位估计结果模型。研究从勘探线方向、垂直勘探线方向和竖直方向分别计算变差函数,对球状模型、指数模型、高斯模型的变差函数拟合效果进行了优选,结果表明球型模型拟合效果最好。针对序贯模拟和克里格品位估值效果进行了分析,结果显示:序贯高斯模拟结果在品位分布形态上更接近样品品位分布形态,其平滑效应更小;克里格方法估计与序贯高斯模拟方法相比仅在品位均值方面更接近样本品位均值。因此,认为序贯高斯模拟方法可以更好地刻画矿体内品位分布状态。     

基于地质统计学的NDVI图像估值技术

将疏采样后的NDVI图像作为未受云层影响的已知数据,分别用普通克里格、泛克里格、指示克里格和序贯指示模拟对NDVI图像进行恢复并比较其效果。研究发现,各种克里格法对NDVI图像的估值效果由高到低依次为泛克里格、普通克里格、指示克里格,通常计算方便的普通克里格法就能够满足图像恢复所要求的精度;普通克里格方差和泛克里格方差只能反映数据的构型,不能很好地衡量估值图像的不确定性,指示克里格的条件方差的分布和实际误差的分布基本一致,能够较好地衡量估值图像的不确定性,并且其大小与NDVI影像数据的不确定性大小的分布一致。序贯指示模拟得到的多个等概率实现表现出很大的空间变异性,多个实现的均值图像光滑效应明显,估值精度不高,但是多个实现的方差分布可以很好地表征空间数据的不确定性分布。     

济南泉域岩溶裂隙介质的多重指示克里格法研究

本文选择研究程度较高,具有代表性的山东济南泉域的含水岩溶裂隙介质,运用多重指示克里格方法研究岩溶裂隙介质的空隙连通性能,并预测岩溶裂隙密集带和强径流带的位置。研究表明,多重指示克里格方法能够通过不同边界域值的设定,有效地克服由于非平稳区域化变量产生的参数"漂移",取得较理想的模拟预测效果。     

基于不同地质统计方法的渗透系数场对污染物运移的影响

渗透系数场的空间变异性是影响污染物运移结果的决定因素,而地质统计方法是解决渗透系数空间变异性的主要技术手段。本文利用野外场地实测数据,采用普通克里格法和指示克里格法、顺序高斯模拟法和顺序指示模拟法四种地质统计方法,插值估测和模拟再现随机渗透系数场,进而对比研究四种渗透系数场对大尺度污染物运移的影响。研究结果表明,污染羽的质心位置（一阶矩）主要由渗透系数的平均值来决定;污染羽在空间上的展布范围（二阶矩）主要受渗透系数空间变异方差的影响;条件模拟克服了估计法的平滑效果,较好地再现真实曲线的波动性,渗透系数（ lnK）估计方差与污染羽空间二阶矩随着条件模拟次数的增加而减小,并且顺序指示模拟程度更加明显。     

干旱区剖面土壤盐分空间变异特征及随机模拟

利用便携式土壤盐分计测定干旱区膜下滴灌棉田及裸地剖面土壤电导率,利用GS+软件确定其半方差函数,并进行序贯高斯模拟来研究土壤剖面盐分的空间变异特征。结果显示:干旱区剖面上土壤盐分空间变异性强,膜下滴灌棉田符合球形模型,裸地区符合高斯模型,且棉田变程远小于裸地;棉田埋深50cm以浅土壤盐分变化剧烈,埋深50cm以深变化平缓,在垂向上呈现3个高值区(30,50和75cm深度附近)及3个低值区(埋深40,60和100cm附近);克里格法具有明显的平滑作用,降低了土壤盐分的空间变异性,而序贯高斯模拟数据更离散,更能反映土壤盐分的空间变异性,但多次模拟均值随次数增加趋于稳定。     

尕斯油田岩性建模算法优选

比较岩性模型建立方法。首先,在高分辨率层序地层学的指导下,最大限度地应用地质、露头、三维地震、测井等静态资料,发挥井点资料垂向分辨率高,地震资料横向信息丰富的优势,在地质规律约束下建立不同时间的高精度等时地层格架模型。然后,在精细格架模型的基础上,以测井解释得到的岩相数据作为条件数据,分别采用指示克里格、截断高斯模拟、Object-modeling算法、贯指示模拟建立砂体展布模型。最后,通过抽稀检验评价不同算法对模拟结果的影响,实现算法及其参数的优选,从而指导整个区块不同开发阶段,不同井网密度时全区三维精细地质模型的建立,也可为具有相似地质环境的油田建立三维地质模型提供参考。通过比较,优选出指示克里格、序贯指示模拟两种算法都能较好表征本研究区地质情况。     

山西沁水寿阳ST区块煤储层三维精细地质建模

煤储层三维地质模型的精确性直接影响到后期煤层气开发方案的部署和煤层气井的产量。本文以山西沁水盆地寿阳ST区块为例,基于地质数据、岩心数据、测井数据和地震数据等资料,提出了井震约束条件下煤储层的三维地质建模方法。通过建立构造模型,采用序贯指示模拟方法模拟煤层在三维空间的分布,建立研究区岩相模型。通过序贯高斯模拟方法模拟煤层含气量、孔隙度、渗透率等参数分布规律,建立反映煤层气特征的精细三维属性模型,预测了相关属性参数的空间分布特征。基于地质模型划分了产能潜力区,结合产能数值模拟技术,进行了煤层气井单井产能预测。本文划分的煤层气产能潜力区与产能预测结果,与目前区块内煤层气开发部署和实际产气情况吻合。     

序贯高斯模拟在西藏尼木岗讲矿区的应用研究

利用Sgems和Micromine软件,应用序贯高斯模拟方法对西藏尼木岗讲铜多金属矿北区铜矿体的变化性进行了研究。首先根据钻孔柱状图整理出了矿区的钻孔数据,建立了钻孔数据库和钻孔三维模型,然后对钻孔中的样品分析数据进行了正态性分析,分析得出数据基本服从对数正态分布。对数据进行了标准正态变换,计算了变差函数,利用序贯高斯模拟方法对品位数据进行了10次模拟。最后根据模拟结果分析了矿体的变化性。矿体整体为东西走向,向南缓倾,但钻孔间矿体形态不定,可以有多种连接方式,多次结果的平均值可指示矿体最可能的形态。     

鄂尔多斯盆地西峰油田白马南特低渗岩性油藏储层地质建模

以鄂尔多斯盆地西峰油田白马南长81砂岩油藏为例，应用随机序贯高斯模拟法（SUS）和序贯指示模拟法（SIS）建立研究目标的储层地质模型，进而探讨了在特低渗岩性油藏中相对高渗透储层形成的控制因素。认为长81油藏储层的形成主要受控于沉积作用；成岩作用对改善储层物性及油气的运聚起了重要的作用。砂体的三维模拟结果显示主砂体位于研究区西部，呈西南-东北向展布。储层物性参数的三维模拟也表明优质储层主要分布在主砂体带上长石溶蚀相和绿泥石薄膜胶结成岩相区。     

工程场地的断层密度地质统计学分析

为了深入研究某工程场地内断层的地质特征,通过收集、整理工程场地的地质资料和较系统的野外地质调查,应用指示克里格方法分析了场地内断层密度的空间分布特征。研究表明断层密度具有明显的各向异性分布特征。断层密度分布的主变程方向为296.4°,与工程场地长轴方向一致,在此方向上断层密度具有较好的相关性与连续性,工程地质条件较差。次变程方向为26.4°,在此方向上断层密度的相关性差,工程地质条件则较好。建立了工程场地地表断层密度的定量数学模型,主变程1 692m,次变程759m,主变程方向296.4°,块金值0.127,基台值0.263。划分了场地工程地质带,工程场地南部边缘地质条件较差,中部至北部边缘地质条件相对较好。     

Incorporation of fracture directions into 3D geostatistical methods for a rock fracture system

Simulating a rock fracture distribution is an important problem common to various fields in geosciences. This paper presents GEOFRAC, a geostatistical method to simulate a fracture distribution by incorporating the directions (strikes and dips) of the sampled fracture data into the simulation. Fracture locations are generated randomly following fracture densities assigned by a sequential Gaussian simulation. Fracture directions are transformed into an indicator set consisting of several binary (0 and 1) variables and the variables are compressed using the principal component analysis. Ordinary kriging is then employed to estimate the distributions of these principal values and the results are back-transformed into the coordinate system of the original indicator set. Fracture directions are generated randomly using their histograms within the defined directional interval. Finally, facets (fracture elements) are determined from the simulated locations and directions, and the fractures within the angle and distance tolerances are connected to form a fracture plane. From a case study of applying GEOFRAC to the fracture data in Kikuma granite, southwest Japan, GEOFRAC was shown to be able to depict a plausible fracture system because the simulated directions corresponded well to those measured. Furthermore, the simulated fracture system was available to estimate the hydraulic conductivity of the study site, which was roughly in agreement with the average of hydraulic test results.     

Geostatistical downscaling of fracture surface topography accounting for local roughness

This study proposes a new geostatistical methodology that accounts for roughness characteristics when downscaling fracture surface topography. In the proposed approach, the small-scale fracture surface roughness is described using a “local roughness pattern” that indicates the relative height of a location compared to its surrounding locations, while the large-scale roughness is considered using the surface semivariogram. By accounting for both components–the minimization of the local error variance and the reproduction of the local roughness characteristics–into the objective function of simulated annealing, the fracture surface topography downscaling process was improved compared to standard geostatistical methodologies such as ordinary kriging and sequential Gaussian simulation. Downscaled topography data were then assessed in terms of prediction errors and roughness distribution.     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains

The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.     

Two Ordinary Kriging Approaches to Predicting Block Grade Distributions

Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation.     

Geostatistical approach for identification of transmissivity structure at Dulliu area in Taiwan

 A thorough understanding of the characteristics of transmissivity makes groundwater deterministic models more accurate. These transmissivity data characteristics occasionally possess a complicated spatial variation over an investigated site. This study presents both geostatistical estimation and conditional simulation methods to generate spatial transmissivity maps. The measured transmissivity data from the Dulliu area in Yun-Lin county, Taiwan, is used as the case study. The spatial transmissivity maps are simulated by using sequential Gaussian simulation (SGS), and estimated by using natural log ordinary kriging and ordinary kriging. Estimation and simulation results indicate that SGS can reproduce the spatial structure of the investigated data. Furthermore, displaying a low spatial variability does not allow the ordinary kriging and natural log kriging estimates to fit the spatial structure and small-scale variation for the investigated data. The maps of kriging estimates are smoother than those of other simulations. A SGS with multiple realizations has significant advantages over ordinary kriging and even natural log kriging techniques at a site with a high variation in investigated data. These results are displayed in geographic information systems (GIS) as basic information for further groundwater study. Received: 27 August 1999 · Accepted: 22 February 2000     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Estimation or simulation? That is the question

The issue of smoothing in kriging has been addressed either by estimation or simulation. The solution via estimation calls for postprocessing kriging estimates in order to correct the smoothing effect. Stochastic simulation provides equiprobable images presenting no smoothing and reproducing the covariance model. Consequently, these images reproduce both the sample histogram and the sample semivariogram. However, there is still a problem, which is the lack of local accuracy of simulated images. In this paper, a postprocessing algorithm for correcting the smoothing effect of ordinary kriging estimates is compared with sequential Gaussian simulation realizations. Based on samples drawn from exhaustive data sets, the postprocessing algorithm is shown to be superior to any individual simulation realization yet, at the expense of providing one deterministic estimate of the random function.